Linking Across Data Granularity: Fitting Multivariate Hawkes Processes to Partially Interval-Censored Data

TL;DR

This study introduces the partially censored multivariate Hawkes process (PCMHP), a novel point process that shares parameter equivalence with the MHP and can effectively model both timestamped and interval-censored data.

Abstract

The multivariate Hawkes process (MHP) is widely used for analyzing data streams that interact with each other, where events generate new events within their own dimension (via self-excitation) or across different dimensions (via cross-excitation). However, in certain applications, the timestamps of individual events in some dimensions are unobservable, and only event counts within intervals are known, referred to as partially interval-censored data. The MHP is unsuitable for handling such data since its estimation requires event timestamps. In this study, we introduce the Partially Censored Multivariate Hawkes Process (PCMHP), a novel point process which shares parameter equivalence with the MHP and can effectively model both timestamped and interval-censored data. We demonstrate the capabilities of the PCMHP using synthetic and real-world datasets. Firstly, we illustrate that the PCMHP can approximate MHP parameters and recover the spectral radius using synthetic event histories. Next, we assess the performance of the PCMHP in predicting YouTube popularity and find that the PCMHP outperforms the popularity estimation algorithm Hawkes Intensity Process (HIP). Comparing with the fully interval-censored HIP, we show that the PCMHP improves prediction performance by accounting for point process dimensions, particularly when there exist significant cross-dimension interactions. Lastly, we leverage the PCMHP to gain qualitative insights from a dataset comprising daily COVID-19 case counts from multiple countries and COVID-19-related news articles. By clustering the PCMHP-modeled countries, we unveil hidden interaction patterns between occurrences of COVID-19 cases and news reporting.

Figures (7)

• Figure 1: Example of multi-platform interaction between view events on YouTube (red lollipops) and tweets on Twitter (blue lollipops). The data is partially interval-censored, as YouTube does not expose individual views, but only the view counts $\mathsf{C}_i$'s over the predefined intervals $[o_i, o_{i+1})$ (shown as red rectangles). The dashed lines show the latent branching structure between views and tweets. The red lollipops are also dashed and empty, indicating that YouTube views are not observed.
• Figure 2: Comparison of performance metrics in the parameter recovery experiment across model fits: MHP (i.e. the data-generating process), PCMHP-PP and PCMHP-IC for varying interval sizes (1, 2, 5, 10 and 20). (left to right) RMSE for each parameter type $\{\boldsymbol{\alpha}, \boldsymbol{\theta}, \boldsymbol{\nu}\}$ and spectral radius estimation error $\Delta \rho$. Samples are drawn from a 2-dimensional MHP with spectral radius $\rho(\boldsymbol{\alpha}) = 0.75$. Each boxplot represents the distribution of errors calculated from 50 estimates of $\boldsymbol{\Theta}$, each obtained from a different group in the synthetic dataset. The mean and median estimates are indicated by the dashed green lines and solid orange lines, respectively.
• Figure 3: (Left) Relating the spectral radius estimation error $\Delta \rho$ of $\textsc{PCMHP}\xspace(5, e)$ and the number of MBP dimensions $e$. Note that $\textsc{PCMHP}\xspace(5, 0)$ is the MHP (i.e. the data-generating process). (Right) Relating the spectral radius estimation error $\Delta \rho$ of $\textsc{PCMHP}\xspace(d, 1)$ and the model dimensionality $d$. In both plots, samples are drawn from a $d$-dimensional MHP with spectral radius $\rho(\boldsymbol{\alpha}) = 0.92$. Hyperparameters are $T=100$, $N_{sequences}=20$ and intervalsize=$1$. We fit two models for each $\textsc{PCMHP}\xspace$ column: $\textsc{PCMHP}\xspace-\text{PP}$ (i.e.$\textsc{PCMHP}\xspace$ fit on timestamp data on all dimensions) and $\textsc{PCMHP}\xspace-\text{IC}$ (i.e.$\textsc{PCMHP}\xspace$ fit on interval-censored data on the first $e$ dimensions and timestamp data on the last $d-e$ dimensions). The mean and median estimates are indicated by the dashed green lines and solid orange lines, respectively.
• Figure 4: Comparison of fits and predictions of our proposal $\textsc{PCMHP}\xspace(3,2)$ and the baseline HIP Rizoiu2017 for views (left), shares (center) and tweets (right) for a sample video from ACTIVE: a trailer for the 2014 movie Whiplash (id $7d\_jQycdQGo$). The first 90 days are used to fit model parameters, while the next 30 days (indicated by the gray shaded area) are unseen by the model and used for evaluation. HIP does not predict the share and tweet counts, as it treats these as exogenous inputs. The blue shaded area shows prediction uncertainty computed for the $\textsc{PCMHP}\xspace(3,2)$ fits.
• Figure 5: Performance comparison of $\textsc{PCMHP}\xspace(1,1)$, $\textsc{PCMHP}\xspace(2,2)$, $\textsc{PCMHP}\xspace(2,1)$-jitter and $\textsc{PCMHP}\xspace(2,1)$ on the COVID case count prediction task over our sample of 11 countries. The dashed line and solid line indicate the mean and median estimates, respectively.

Theorems & Definitions (4)

• Definition 1
• Theorem 2
• Theorem 3
• Theorem 4